Every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra

TL;DR

This paper proves that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra, using a new graph construction called the porcupine graph, which ensures the isomorphism respects grading.

Contribution

It introduces the porcupine graph construction to establish graded isomorphisms between ideals and Leavitt path algebras, refining previous isomorphism results.

Findings

Every graded ideal is graded isomorphic to a Leavitt path algebra.

The porcupine graph provides a graded isomorphism framework.

Extension to gauge-invariant ideals in graph C*-algebras.

Abstract

We show that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. It is known that a graded ideal $I$ of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the generalized hedgehog graph, which is defined based on certain sets of vertices uniquely determined by $I$. However, this isomorphism may not be graded. We show that replacing the short "spines" of the generalized hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) such that its Leavitt path algebra is graded isomorphic to $I$. Our proof adapts to show that for every closed gauge-invariant ideal $J$ of a graph $C^*$-algebra, there is a gauge-invariant $*$-isomorphism mapping the graph $C^*$-algebra of the porcupine graph of $J$ onto $J.$